%I #24 Dec 17 2023 08:57:22
%S 9,5,4,9,1,8,7,8,9,9,8,7,6,7,4,1,0,3,7,5,1,2,3,3,9,7,8,1,1,0,2,9,1,0,
%T 7,7,6,3,2,7,1,5,3,7,3,8,0,7,8,0,5,2,8,3,1,4,8,7,9,9,1,9,1,6,7,6,0,9,
%U 4,0,3,5,6,8,6,7,1,4,5,3,9,5,3,4,9,8,1,5,1,8,6,7,4,4,6,1,0,9,8,7,6,7,4,9
%N Decimal expansion of phi(exp(-Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.
%H Istvan Mezo, <a href="http://arxiv.org/pdf/1106.2703.pdf">Several special values of Jacobi theta functions</a> arXiv:1106.2703v3 [math.CA] 24 Sep 2013
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler_function">Euler function</a>
%F phi(q) = QPochhammer(q,q) = (q;q)_infinity.
%F phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
%F phi(exp(-Pi)) = exp(Pi/24)*Gamma(1/4)/(2^(7/8)*Pi^(3/4)).
%e 0.954918789987674103751233978110291077632715373807805283148799191676094...
%t phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi]], 10, 104] // First
%Y Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).
%Y Cf. A292820, A292824, A292828.
%K nonn,cons,easy
%O 0,1
%A _Jean-François Alcover_, Jun 19 2015